Mar 20, 2007 - According to the inflationary scenario, all structure in the Universe originates ..... and apparatus alone, without taking into account the quantum ...
POINTER STATES FOR PRIMORDIAL FLUCTUATIONS IN INFLATIONARY COSMOLOGY
arXiv:astro-ph/0610700v2 20 Mar 2007
Claus Kiefer and Ingo Lohmar Institut f¨ ur Theoretische Physik, Universit¨at zu K¨oln, Z¨ ulpicher Str. 77, 50937 K¨oln, Germany. David Polarski Laboratoire de Physique Th´eorique et Astroparticules, UMR 5207 CNRS, Universit´e de Montpellier II, 34095 Montpellier, France. Alexei A. Starobinsky Landau Institute for Theoretical Physics, Kosygina St. 2, Moscow 119334, Russia.
Abstract Primordial fluctuations in inflationary cosmology acquire classical properties through decoherence when their wavelengths become larger than the Hubble scale. Although decoherence is effective, it is not complete, so a significant part of primordial correlations remains up to the present moment. We address the issue of the pointer states which provide a classical basis for the fluctuations with respect to the influence by an environment (other fields). Applying methods from the quantum theory of open systems (the Lindblad equation), we show that this basis is given by narrow Gaussians that approximate eigenstates of field amplitudes. We calculate both the von Neumann and linear entropy of the fluctuations. Their ratio to the maximal entropy per field mode defines a degree of partial decoherence in the entropy sense. We also determine the time of partial decoherence making the Wigner function positive everywhere which, for super-Hubble modes during inflation, is virtually independent of coupling to the environment and is only slightly larger than the Hubble time. On the other
hand, assuming a representative environment (a photon bath), the decoherence time for sub-Hubble modes is finite only if some real dissipation exists.
2
1
Introduction
According to the inflationary scenario, all structure in the Universe originates from quantum vacuum fluctuations during a de Sitter (inflationary) stage in the very early Universe. These are inhomogeneous fluctuations (perturbations) of both the space-time metric and a scalar inflaton field. While the tensor part of the metric fluctuations generated during inflation produces the primordial gravitational wave background [1], its scalar part together with perturbations of the inflaton leads to the origin of primordial density fluctuations producing present gravitationally bound objects and the large-scale structure of the Universe [2]. All these fluctuations can be formally represented by a scalar field with a time-dependent ‘mass’, the time dependence coming from the coupling to the expanding universe described by a scale factor a(t) (we assume here a spatially flat Friedmann-Robertson-Walker (FRW) cosmological model). Usually one assumes these fluctuations to be in their ground state (the adiabatic vacuum state) at the onset of inflation. This choice follows from the hypothesis of the maximal possible symmetry of the Universe in some period in the past during inflation. Also, it can be shown that this initial condition for fluctuations is an attractor for a wide open set of other initial conditions with a non-zero measure. The smallness of the fluctuations means that different modes (distinguished by a wave vector k with the wave number k = |k|) decouple and can be treated separately. Modes relevant for structure formation cross the Hubble radius H −1 twice. Here H ≡ a/a ˙ is the Hubble parameter (with the dot denoting d/dt). During inflation, their wavelength λ = 2πa/k becomes bigger than H −1 (‘the first Hubble radius crossing’). The quantum modes then acquire classical properties in the following sense. First, even without considering any interaction with other degrees of freedom, expectation values of any physical quantities constructed from the quantum fluctuations become practically indistinguishable from mean values of corresponding classical quantities as functions of classical stochastic fluctuations. This is achieved due to huge squeezing (in the sense of quantum optics) of those modes that gives a possibility to neglect non-commuting parts of all mode quantum operators. Since no consideration of environmental degrees of freedom and their interaction with modes involved are needed for this indistinguishability, this quantum-to-classical transition was called ‘decoherence without decoherence’ [3]. This result can be generalized to all higher non-linear orders of metric perturbations if only the growing (quasi-isotropic) mode of perturbations is kept [4]. In this way, it is possible to explain the classical stochastic behaviour of observed cosmological density fluctuations including their power spectrum and statistics. However, more subtle ques1
tions like the correct calculation of the entropy of primordial fluctuations cannot be solved in this approximation. Second, this classical behaviour is preserved, and even re-enforced, in the presence of an ‘environment’. Since such an environment is always present in the form of other (‘irrelevant’) fields and fluctuations (or non-linear couplings of the same field), its influence has to be taken into account. In laboratory, it is usually the environment that leads to the emergence of classical behaviour, a process which is called decoherence [5]. In particular, strongly squeezed quantum states are known to be especially sensitive to decoherence. We have demonstrated some time ago that the classical basis distinguished by the environment (the ‘pointer basis’) is – in the limit of large squeezing – approximately given by the basis of field amplitudes [6] (see also [7] for a detailed conceptual discussion). The correct choice of the pointer basis is of particular relevance for the entropy of primordial fluctuations. We have calculated the entropy for each mode and found that it can assume at most half of its maximal value before the second Hubble radius crossing [8]. The maximal entropy would be achieved if the pointer basis was the particlenumber basis [9]. The wavelength of the modes becomes again smaller than H −1 (‘the second Hubble radius crossing’) during the radiation and matter dominated phases. The scalar modes have left their imprint on the anisotropy spectrum of the cosmic background radiation, where they can be observed in the form of acoustic peaks (see e.g. [10, 11] for recent observations). Oscillations of the same origin (the ‘Sakharov oscillations’, often called baryon acoustic oscillations) have been recently discovered in the three dimensional power spectrum of galaxy inhomogeneities [12]. These peaks would be absent if the entropy of each mode assumed its maximal value [8]. This demonstrates that questions of the quantum-to-classical transition for primordial perturbations are not only of academic nature, but have observational relevance. Primordial gravitational waves (tensor modes) generated during inflation also produce primordial peaks in the CMB temperature anisotropy and polarization multipoles (see [13] for discussion of perspectives of their observation). In spite of these investigations, the discussion about the precise mechanism of this quantum-to-classical transition and the amount of entropy of primordial fluctuations gained in its course goes on, cf. the recent papers [14, 15, 16, 17, 18, 19]. It was argued, for example, that the pointer basis is not given by the field-amplitude basis, but by the basis of two-mode coherent states [14], by the squeezed-state basis [18], or by the second-order adiabatic basis [20]. It was further argued in [14] that the associated entropy of fluctuations (being half the maximal entropy) provides a lower (instead of upper) bound on the entropy. Also, there is a disagreement between [18] and 2
[19] regarding the question whether decoherence may occur already during inflation (after the first Hubble radius crossing, of course) or only after its end, during reheating. We find it therefore appropriate to address this issue again by reviewing and extending our line of arguments. Our article is organized as follows. In Section 2 we first state the problem of the quantum-to-classical transition and briefly review the central general concepts. We then apply these concepts to the primordial fluctuations. We give exact and approximate expressions for the von Neumann entropy and also calculate the linear entropy. We then discuss why the pointer states for the fluctuations is given by approximate field-amplitude eigenstates (narrow Gaussians) and not by another basis such as the coherent-state basis. In Section 3 we apply methods from open-system quantum theory (the Lindblad equation) to calculate in a general setting decoherence times for modes outside and inside the Hubble radius. We conclude from this again that the pointer states are narrow Gaussians in the field amplitude. The entropy per mode is thus bounded from above by half the maximal entropy in the long-wave regime before the second Hubble radius crossing.
2 2.1
Quantum-to-classical transition and entropy Quantum description for the primordial fluctuations
The primordial fluctuations which lead to the observed structure in the Universe can originate from the quantum fluctuations of the metric and a scalar field in an early phase where the Universe is expanding in a (quasi-) exponential way (‘inflation’). These fluctuations are assumed to be small so that their mutual interaction can be neglected in the description of the main scenario. This linear treatment of the fluctuations is certainly justified by the smallness of the observed CMB anisotropy and of the ratio of the modulus of the Newtonian potential to the square of the light velocity for all observed gravitationally bound structures in the Universe (apart from the very close vicinity of black holes). It is thus appropriate to represent the modes in Fourier spaces where the various modes (wave numbers) decouple form each other. The dynamics of both the tensor and the scalar modes can be represented by some scalar field φk . Because of the mode decoupling we shall in the following skip the index k. It turns out to be convenient to work with the rescaled field amplitude y ≡ aφ. In the case of tensor fluctuations (gravitational waves), this is already the appropriate variable to deal with; for scalar perturbations one has to use a gauge-invariant combination of metric 3
and inflaton perturbations. These details are irrelevant for the study of the quantum-to-classical transition. Since we are working in Fourier space, y is complex. This reflects the fact that we shall have a two-modes state, the two modes being given by k and −k. For the following discussion it is, however, sufficient to assume that y represents one real mode, but one should keep in mind that the actual number of degrees of freedom is twice as much. This also holds for the entropy discussed below; the entropy for the two-mode state is twice as much as for the one mode considered here. The dynamics is then described by the Hamilton operator 1 2 2a′ 2 2 ˆ H= p +k y + yp , (1) 2 a where a′ ≡ da/dη, with η denoting conformal time, and p is the momentum canonically conjugate to y. (We have skipped the index k.) This Hamiltonian follows after expanding the metric and the scalar field up to second order in the inhomogeneities, with the cosmological background given by a flat Friedmann-Lemaˆıtre universe. Each mode satisfies the Schr¨odinger equation i~
∂ψ(y, η) ˆ = Hψ(y, η) . ∂η
(2)
Assuming that the modes are initially (at the onset of inflation) in their ground state, their wave function is then at any time of the Gaussian form, ψ(y, η) =
2ΩR (η) π
1/4
exp −Ω(η)y 2 ,
Ω ≡ ΩR + iΩI .
(3)
Non-Gaussian initial states can also be studied [21]; it was shown there that there is a wide class of initial non-vacuum states that cannot be distinguished from a vacuum initial state by just looking at the statistics of observable quantities. One can prove from the form of the Hamiltonian (1) that the states (3) represent squeezed states; the squeezing is generated by the last term in (1). (Actually, we have a two-modes squeezed state, but for simplicity we present one mode only, cf. the remark above.) Introducing the squeezing parameter r and the squeezing angle ϕ, one can write (setting ~ = 1 from now on) ΩR =
k , cosh 2r + cos 2ϕ sinh 2r 4
ΩI = −ΩR sin 2ϕ sinh 2r ,
(4)
with ΩR = k and ΩI = 0 for the initial vacuum state. Both r and ϕ are functions of t (resp. η), the exact dependence arising from the expansion law a(t). For pure exponential inflation described by a(t) = a0 exp(HI t) = −
1 , (η − 2ηe )HI
where ηe denotes the conformal time at the end of inflation, simple analytic results are available corresponding to the adiabatic vacuum in the de Sitter space-time [3, 22]: sinh r =
aHI , 2k
cos 2ϕ = tanh r .
(5)
This would lead to r → ∞, ϕ → 0 if inflation lasted arbitrarily long. Inflation, however, ends, but it still leads to r ≈ 120 for the largest cosmological states [23], so one can certainly neglect after some time terms of the order of exp(−r) (‘decaying mode’). From (4) one then finds the asymptotic values ΩR → ke−2r ,
ΩI → −ke−r ,
(6)
exhibiting ΩI ≫ ΩR (WKB limit). This behaviour is assumed to occur generically for an inflationary model. One recognizes that the width of the Gaussian (3) becomes very broad in y and highly squeezed in p. This broadness reflects the fact that the kinetic term of the Hamiltonian becomes negligible in this limit; in the Heisenberg picture this is exhibited by the fact that the kiˆ kin , then approximately commutes with the netic term of the Hamiltonian, H field-amplitude operator, yˆ. In the limit of high squeezing, quantum expectation values become indistinguishable from mean values of classical stochastic quantities, the difference containing typically terms of the order exp(−r) or powers thereof. In a quasi de Sitter stage, the annihilation operator evolves according to aHI a(k) → a(k)0 − a† (−k)0 (7) 2k for k ≪ aH. So far, the primordial fluctuations behave fully quantum. They are described by states which are very broad wave packets. Such broad packets would exhibit in a laboratory situation very non-classical behaviour, as could be checked in interference experiments. The standard scenario of structure formation starts, however, from classical stochastic fluctuations. A major issue is thus to describe the quantum-to-classical transition for the primordial modes. We shall now first briefly review the general concepts and subsequently shall apply them to understand the emergence of classical behaviour for these modes. 5
2.2
General concepts of the quantum-to-classical transition
The superposition principle is at the heart of quantum theory. As a consequence, the set of classical states is of measure zero, since one can always superpose different classical states (e.g. different narrow wave packets) to get ‘weird’ non-classical states. Such states may occur in many ordinary measurement situations. If we assume that a measured system is initially in the state |ni and the measurement device in some initial state |Φ0 i, the evolution according to the Schr¨odinger equation in the simplest case reads t
|ni|Φ0 i −→ exp (−iHint t) |ni|Φ0 i = |ni|Φn (t)i ,
(8)
where Hint is a special interaction Hamiltonian (assumed here to dominate over the free Hamiltonians) which correlates the system state with the device without changing the former. The resulting apparatus states |Φn (t)i are often called ‘pointer states’. In order to yield distinguishable outcomes, these pointer states must be approximately orthogonal. A process analogous to (8) can also be formulated in classical physics. The essential new quantum features now come into play when one considers a superposition of different eigenstates (of the measured ‘observable’) as the initial state. The linearity of time evolution immediately leads to ! X X t cn |ni |Φ0 i −→ cn |ni|Φn (t)i . (9) n
n
But this state is a superposition of macroscopic measurement results. It is by now well established, both theoretically and experimentally, that the ubiquitous and unavoidable interaction with the environment has to be taken into account [5]. The measurement device is itself ‘measured’ (passively recognized) by the environment according to ! X X t cn |ni|Φn i |E0 i −→ cn |ni|Φn i|En i. (10) n
n
This is again a macroscopic superposition, now including the myriads of degrees of freedom pertaining to the environment (gas molecules, photons, etc.). However, most of these environmental degrees of freedom are inaccessible. Therefore, they have to be integrated out from the full state (10). This leads to the reduced density matrix for system plus apparatus, which 6
contains all the information that is available there. It reads X ρSA ≈ |cn |2 |nihn| ⊗ |Φn ihΦn | if hEn |Em i ≈ δnm ,
(11)
n
since under realistic conditions, different environmental states are orthogonal to each other (they can discriminate between different states of the apparatus). Equation (11) is identical to the density matrix of an ensemble of measurement results |ni|Φn i. System and apparatus thus seem to be in one of the states |ni and |Φn i, given by the probability |cn |2 . This is decoherence. The interaction with the environment singles out a preferred basis for the apparatus, which is just the above-mentioned pointer basis. Assuming the usual Born rule for quantum probabilities, the reduced density matrix contains all the information that can be obtained from system and apparatus alone, without taking into account the quantum entanglement with the environment explicitly. In our case it is the primordial fluctuations for which a classical behaviour has to be obtained. Formally, they correspond to the apparatus states above, but since we have no additional system, we just call them the system variables; they are in interaction with the environment. The latter is represented by other fields or by higher-order modes of the metric and inflaton itself. An important property of a pointer basis is its preservation in time – they should be robust while interacting with the environment, that is, they should not evolve into superpositions of themselves. This basis thus represents the properties that become classical. For this, two conditions have to be fulfilled [5]: first, the projection operators on the pointer states should approximately commute with the interaction Hamiltonian between system and environment; second, these projection operators should approximately commute at different times (in quantum optics, this is referred to as a quantum non-demolition condition). We shall see in the following subsections that these conditions are fulfilled in the high-squeezing limit for the field-amplitude basis of the primordial fluctuations. In the literature, various methods are discussed with which one can determine the pointer basis [5]. It has been shown that they are equivalent to each other at least in simple situations [24]. One prominent method makes use of the entropy of the system. As long as it is in a pure state, this entropy is zero. If it is in a mixed state due to decoherence, this entropy is positive. Its value characterizes the degree of entanglement between system and environment – the stronger the entanglement, the higher the entropy. One can thus find the most robust system states (that is, the pointer states) by minimizing the resulting entanglement entropy. This method is called the ‘predictability sieve’ [25]. We shall apply this method below to the primordial fluctuations. 7
A final tool for the study of the quantum-to-classical transition should be introduced – the Wigner function. For a general density matrix in the position representation, ρ(x, x′ ), it is defined by the expression Z 1 ∞ W (x, p) = dy e2ipy ρ(x − y, x + y) . (12) π −∞ It is thus defined on phase space where it gives the correlations between position and momentum. It is not, in general, positive definite, reflecting the fact that quantum theory is inequivalent to a classical statistical theory on phase space.
2.3
Wigner function and reduced density matrix for the primordial fluctuations
We shall first address the system of primordial fluctuations as if it were isolated [3]. In order to recognize correlations between position (here: field amplitude) and momentum (here: the variable canonically conjugate to the field amplitude), the Wigner function can be calculated. In general it can assume negative values, but for the special case of a Gaussian wave function it is everywhere positive (of a Gaussian form). This is the case here where the fluctuations are assumed to be in their ground state. A convenient quantity is then the ‘Wigner ellipse’ defined by the contour of this Gaussian Wigner function, because it provides one directly with a measure for the mean square deviations of y and p. From the analysis of the Wigner function one can see that the system behaves as if it had a stochastic amplitude (with a Gaussian distribution if the state is (3)) and a fixed phase, see the calculations in [3, 6, 7]. This phase is given by the squeezing angle. The situation here is thus already a peculiar one since the quantum nature of the squeezed state cannot be seen, in the high-squeezing limit, from expectation values and variances [3]. The scenario for the isolated system of primordial fluctuations proceeds as follows. After the first horizon crossing (during inflation) one has r . 100, ϕ → 0, and the Wigner ellipse will have – after an appropriate re-scaling of the axes – a major half axis α = exp(r) and a minor half axis β = exp(−r). After the second horizon crossing (in the radiation or matter dominated phase) the Wigner ellipse rotates, with small oscillations around a large mean value of r. The rotation is very slow, the frequency being about the inverse age of the universe, which is why the squeezed nature is preserved for a long time [26].1 For modes that re-enter in the radiation era, this rotation leads to 1
In contrast to this, the rotation is fast in the case of black holes that has important
8
the acoustic peaks in the cosmic background radiation [3, 22]. These peaks have been observed with high precision [10, 11]. This discussion treats fluctuations as being isolated, which is, however, unrealistic [5, 28, 29]. Other fields (the ‘environment’) interact with them, even if the coupling is weak. Such a coupling may also arise from a small self-interaction of the modes [15]. Even if there were no other fields present, this small self-interaction would remain. As long as the modes are outside the Hubble radius, this coupling cannot lead to a direct causal interaction, but can only produce entanglement (‘EPR-type situation’). Since usual interactions couple to fields (instead of canonical momentum), the coupling is in y (not p). This already suggests that the y-basis is equal to the pointer basis at least approximately. Spatial gradients would not change this conclusion, since they do not depend on p. The situation is analogous to the localization of particles in quantum mechanics [5, 30]. This interaction with the environment without direct causal contact can be described by multiplication of the density matrix ρ0 (y, y ′) corresponding to the system alone with a Gaussian factor according to ξ ′ ′ ′ ′ 2 ρ0 (y, y ) −→ ρξ (y, y ) = ρ0 (y, y ) exp − (y − y ) , (13) 2 cf. [8, 29]. The parameter ξ encodes phenomenologically the details of the interaction strength with the environment. This interaction is by no means restricted to be linear and can be very complicated. In this way, non-diagonal elements are suppressed with respect to the y-basis. In a realistic situation for decoherence, one would expect that ξ dominates over the corresponding part in (3), ξ ≫ ΩR ≈ ke−2r . (14)
The typical time scale for decoherence during inflation is td ∼ HI−1 . (There is a close analogy to chaotic systems, with HI corresponding to the Lyapunov parameter [8]. The reason for this is the classical instability of our system.) From analogous investigations in quantum mechanics [31], it can realistically be expected that the parameter ξ settles to a constant value after a certain transition time. The axes of the Wigner ellipse then read r ξ α ≈ er , β ≈ ≫ e−r , (15) k where (14) has been used. While the major axis has remained unchanged, the minor axis has become bigger and settles to a constant value. The area consequences for Hawking radiation [27].
9
of the Wigner ellipse thus increases, leading to a non-vanishing entropy (see below). In order to avoid conflict with observation, one has to impose the ‘correlation condition’ β ≪ α, leading to ξ ≪ e2r . k
(16)
Together with the slow rotation of the ellipse, this guarantees the formation of acoustic oscillations. A random distribution of p and q would totally smear out these structures.
2.4
Entropy
As long as the fluctuations are treated as an isolated system, they can be described by a wave function and thus have vanishing entropy. This is no longer the case for the interacting system described by (13); the entanglement with the environment leads to a ‘loss of information’ for the system and therefore to a positive entropy. The quantum correlations are present only in the total system and are therefore unseen ‘locally’. The entropy is calculated from S = −tr(ρξ ln ρξ ) , (17) where we have set kB = 1. The entropy has, of course, also to be calculated for a two-mode squeezed state, although for simplicity we give here again the calculation for one real mode only. For the full entropy per the two-mode state, one thus has to double the results below. It is convenient to introduce the dimensionless parameter χ = ξ/ΩR controlling the strength of decoherence. (In the case of pure exponential inflation one has χ = ξ(1 + 4 sinh2 r)/k.) Inserting (13) into (17), one gets the explicit expression √1 + χ − 1 1 p 2 − S = − ln √ 1 + χ − 1 ln √ 1+χ+1 2 1+χ+1 √ (18) p 1√ 1+χ−1 = ln χ − 1 + χ ln . √ 2 χ One recognizes that the entropy vanishes for ξ → 0, as it must. In the limit χ ≫ 1 (large decoherence) one gets S = 1 − ln 2 +
ln χ + O(χ−1/2 ) . 2
Both (18) and (19) are displayed in Figure 1. 10
(19)
2
S, (18) Approximation, (19)
1.8 1.6 1.4
S
1.2 1 0.8 0.6 0.4 0.2 0 0
5
10
15
20
χ Figure 1: Entropy in dependence on the decoherence parameter χ. Shown are the exact expression and the limiting case. It is seen that the asymptotic value is readily attained. For χ ≈ 1.5 one obtains S ≈ ln 2, corresponding to the loss of one bit of information. Usually decoherence sets in roughly at this stage [5]. A stronger condition than (16) is obtained by the reasonable demand that some squeezing should remain compared to the vacuum state which has ΩR = k. This leads to the condition ξ < k and yields for exponential inflation in the high-squeezing limit the bound S . r. Whether this really holds is, of course, a question about realistic interactions in the universe. The maximal entropy is instead given by Smax = 2r (obtained by smearing out the Wigner ellipse into a big circle and corresponding to the pointer basis being the particle-number basis [9]). One can thus define a notion of partial decoherence in the entropy sense by the ratio of entropy to maximal entropy, S/Smax , per each mode. It is also instructive to consider the linear entropy. It is defined by Slin = tr ρ − ρ2 , (20) 11
obeying 0 ≤ Slin < 1. Inserting (13), one obtains Slin = 1 − (1 + χ)−1/2 .
(21)
For χ ≪ 1 one gets Slin ≈ χ/2, while for χ ≫ 1 one gets Slin ≈ 1 − χ−1/2 < 1 − exp(−r), where we have used the bound ξ < k in the last step. As was already mentioned, the fact that the interaction with the environment is in y-space leads to the suggestion that – for large squeezing – the pointer basis equals approximately the field-amplitude basis [6]. Exact diagonalization would correspond to an entropy increase given by S = r = Smax /2 [32], in accordance with the expectation that S < r holds for modes outside the horizon. Contrary to this it was claimed in [14] that the pointer basis is given by (two-mode) coherent states. Diagonalization in such a basis was in the cosmological context first addressed in [33], with the result that S approaches for large squeezing the value r = Smax /2, that is, the same value as for the field-amplitude basis. Figure 2 displays the entropy S(r) for the field-amplitude basis, the coherent-state basis (‘z-basis’), and the particlenumber basis (‘n-basis’). It is easily seen that S approaches r in the first two cases, whereas it approaches 2r in the latter case.
2.5
Pointer basis for the primordial fluctuations
We shall now re-enforce from various points of view our earlier result [6, 7] that the pointer basis is for large squeezing (approximately) given by the field-amplitude basis. The first argument is about the normalizability of the pointer basis. The authors in [14] are concerned about the fact that field-amplitude eigenstates are not normalizable and that they, moreover, exhibit infinite momentum spread. However, even if one dealt with exact amplitude (‘position’) eigenstates, the quantum mechanical formalism would be able to cope with such a situation (GNS construction instead of Hilbert space). But the point is that the environmental parameter ξ entering (13) is never strictly infinite, so that a finite, albeit small, off-diagonal part for the density matrix remains. One is thus dealing with very narrow wave packets that are perfectly normalizable. Even if we had exact position eigenstates, an ‘infinite’ momentum spread would only take place if the kinetic term played a crucial role in the dynamics. As has already been remarked above, however, for the highly squeezed modes the kinetic term is negligible compared to the potential term. This irrelevance of the kinetic term is also the reason why the reference to [25] in [14] is misleading. In [25], the question of the pointer basis has been studied for an ordinary harmonic oscillator – in contrast to cosmology 12
2
Reduction in |ni-basis Reduction in |yi-basis Reduction in |zi-basis
n
S
1.5
1
0.5
y z 0 0
0.2
0.4
0.6
0.8
1
r Figure 2: Entropy S in dependence of the squeezing parameter r for three cases of the pointer basis.
13
where one deals more with an upside down oscillator [34] – coupled to a high-temperature environment, using the method of the ‘predictability sieve’ (minimal local entropy production). It was found that this basis is given by the coherent-state basis.2 A crucial ingredient in this proof, however, was the importance of both the kinetic and potential term for the oscillator (an averaging procedure over many oscillation cycles had to be performed). For small times, when the kinetic term does not yet become relevant, it was found in [25] that, in fact, the position basis is the (approximate) pointer basis. This is the limit of relevance for the cosmological fluctuations. In situations where the system Hamiltonian is negligible, the pointer basis has the property that it consists of states which are eigenstates of an operator that commutes with the interaction Hamiltonian. If the system Hamiltonian is not negligible (in our case, this happens for small r), one must invoke a principle such as the ‘predictability sieve’ or the ‘rate of de-separation’ in order to determine a pointer basis. In the case of a free particle coupled to a localizing environment it was shown that the predictability sieve (as well as other criteria) predict robust pointer states that are narrow Gaussian wave packets [24]. In the next section we shall apply methods of open system dynamics to the Hamiltonian (1) in order to discuss the decoherence time for the cosmological fluctuations with wavelengths both bigger and smaller than the Hubble scale. This will enforce our earlier result about the nature of the pointer basis. Another argument follows from the requirement that the pointer basis should be stable with respect to time evolution. In the limit of no interaction with environment, this means that it should be stable with respect to free evolution. However, the coherent state basis does not satisfy this requirement in the period between the two Hubble radius crossings: expansion of the Universe causes coherent states to become strongly squeezed ones in a characteristic time ∼ H −1 . This can be seen as follows. It has been shown [22, 36] that the time evolution generated by any quadratic Hamiltonian like (1) factorizes into an irrelevant phase rotation and the action of a squeezing operator with parameters r and ϕ. It is further well known (e.g. [37] Sect. 2.7, and [36]) that a squeezing operator acting on an arbitrary coherent state |zi (not just the vacuum ground state) produces a ‘squeezed coherent state’: It has characteristic uncertainties proportional to e−r and er along rotated axes in the y-p space, coinciding with the squeezed and the stretched semi-axes of the Wigner ellipse, respectively. If we thus consider the time evolution of a coherent state 2
This result had already been found by another method – implementing the rate with which initially unentangled states become entangled – in [35].
14
starting at a time ti after the mode has left the Hubble scale, we have to be concerned solely with the development of the squeezing parameters r and ϕ starting with r = 0 = ϕ at that time. As in the particular de Sitter example mentioned before, the squeezing angle ϕ then quickly ‘freezes out’ and the squeezing factor grows like r ≈ ln(a/ai ), a result first obtained for gravitational waves [23], and later generalized to scalar perturbations [22]. For quasi-exponential inflation, we consequently have r ≈ HI (t − ti ), and it takes a very short time ≈ HI−1 for the state to become significantly squeezed. But even if we start in the ensuing radiation-dominated era, where a ∝ (t/te )1/2 and H = 1/(2t), we obtain r ≈ ln(t/ti )/2 = ln(2Hi t)/2, so that squeezing still becomes effective at a time t ≈ (e2 /2)Hi−1 = e2 ti (with Hi denoting the Hubble parameter at the initial time ti ). Finally, these considerations also apply to two-mode coherent states (cf. the above references). On the other hand, the field-amplitude basis is stable with respect to time evolution in the super-Hubble regime, since the field amplitude remains constant with great accuracy. Here, the squeezing manifests itself in the continuous decrease of the decaying mode (see e.g. [38]). An interesting and subtle situation arises with the second order adiabatic basis proposed in [20]. In the case of the exactly massless and minimally coupled scalar field (1) and the exact de Sitter background, the vacuum mode functions φ˜k of the second order adiabatic vacuum (as defined in this paper) coincide with the exact solution for φk = yk /a. This has led the authors of [20] to the statement that there is no creation of massless minimally coupled particles in the de Sitter space-time. However, this result is a purely academic one since in case of the stable, eternal de Sitter expansion there is no possibility for an observer to measure these particles because they are beyond his/her future event horizon.3 On the other hand, for a viable inflationary model in which the Hubble parameter H decreases during inflation, the mode functions φ˜k do not coincide with the exact solution for φk which √ is constant far outside the Hubble radius (k ≪ aH) and equal to H(tk )/ 2k 3 there (tk is the moment of the first Hubble radius crossing when k = aH). Really, the second-order adiabatic vacuum basis functions were defined in [20] in the form Z t k 2k 2 1 ′ ′ ˜ exp −i dt Wk (t ) , Wk = · (22) φk = √ 3 a 2k 2 + a2 (H˙ + 2H 2 ) 2a Wk for a spatially flat FRW model with an arbitarty a(t). For a quasi-de Sitter ˙ ≪ H 2 , a(η) ≈ −1/H(η)η (η < 0), this expression (slow-roll) regime |H| 3
So, it is reminiscent of the problem, much discussed in the past, of radiation reaction for a charged particle eternally moving in a constant electric field.
15
reduces to
i H(η) −ikη ˜ −η + φk = √ e . (23) k 2k √ Outside the Hubble radius, |φ˜k | = H(η)/ 2k 3 ∝ H(t). Since H(t) < H(tk ) for t > tk , this means that creation of massless minimally coupled particles during slow-roll inflation does occur even with respect to this basis. By the end of inflation and subsequent heating of matter, we return to standard formulas for the particle number and the energy density power spectrum presented in [1] (for the case of gravitons). Thus, in contrast to the fieldamplitude basis, the second-order adiabatic basis (22) is also not stable with respect to time evolution in the super-Hubble regime even during an inflationary stage with slow-roll. Of course, there exists an ambiguity in the definition of adiabatic vacuum mode functions of a given order (in other words, in the definition of the notion of number of particles in an external varying gravitational field) – alternative definitions may differ by terms of a higher order in the adiabatic parameter aH/k and its time derivatives. However, different variants of adiabatic mode expansion produce essentially the same form of the adiabatic expansion of the average value of the quantum field φ energy-momentum tensor as was shown already in [39, 40]. Note in this connection that the most standard textbook calculation of a next small correction to the WKB solution of the wave equation k2 φ¨k + 3H(t)φ˙ k + 2 φk = 0 (24) a (t) (that corresponds to the Hamiltonian (1)) in the WKB (sub-horizon) regime k ≫ aH) reads Z i 1 −ikη (1 + f ) , f = dt a(H˙ + 2H 2 ) , |f | ≪ 1 . (25) φk = √ e 2k a 2k If we formally use (25) for all k during slow-roll inflation ignoring the condition |f | ≪ 1, it just reduces to the same expression (23), exact for H = const but significantly different from the exact solution for a slowly variable H. See also [41, 42, 43] for recent further improvements of the WKB-expansion in this case, in particular, making it homogeneous in both the sub- and superHubble regimes. We would finally like to address the general criticism of [44] where it is claimed that in the discussion of emerging classical behaviour for the fluctuations a transition from a symmetric quantum state to a non-symmetric classical state is implicitly assumed without justification. The authors of [44] therefore claim that in addition new physics (an explicit wave function 16
collapse) is needed. This is, however, a general issue in quantum theory and plays a role, for example, in spontaneous symmetry breaking, see [45], Sec. 6.1: the initial symmetric state develops into a symmetric superposition of all ‘false vacua’. The standard false vacuum is obtained by selecting one component out of this superposition that can be justified using any interpretation of quantum mechanics, for example the Everett or Copenhagen one, without changing of quantitative predictions of quantum mechanics referring to this component. The same happens here: the initial symmetric vacuum state evolves into a symmetric superposition of inhomogeneous states out of which one component is ‘selected’ [45]. Thus, cosmological perturbations are not specific in this sense and neither solve nor complicate the fundamental problem of the foundations of quantum mechanics.
3 3.1
Decoherence time and pointer states Description of decoherence by master equations
Decoherence is the irreversible emergence of classical properties for a quantum system through its unavoidable interaction with the ‘environment’, that is, with irrelevant degrees of freedom [5]. Ideally, one would solve the Schr¨odinger equation for system plus environment and then trace out the environmental degrees of freedom to obtain the reduced density matrix for the system. This density matrix obeys a master equation which provides a nonunitary and irreversible time evolution. Instead of performing this procedure explicitly, one can directly make a general ansatz for the master equation that can cope with all interesting situations. A convenient form for this equation is the ‘Lindblad form’, cf. Sect. 3.3.2.2 in [5] or Eq. (7) in [46]. The corresponding master equation is Markovian (local in time) and preserves the properties of a density matrix (such as conservation of its trace). Such an equation results from a wide range of realistic interactions; the details of these interactions are encoded in the Lindblad operators. The simplest form is a pure localization term in addition to the system Hamiltonian; such a term leads to the Gaussian suppression of interferences described by (13). The most general interaction may be non-Markovian, but since we are interested in the minimal mechanism to guarantee decoherence, the restriction to the Markovian case is sufficient. In [47], a master equation for the reduced density operator ρˆ was studied for a free particle plus such a localization term. It reads dˆ ρ i D =− [ˆ p2 , ρˆ] − [ˆ x, [ˆ x, ρˆ(t)]] . dt 2m~ 2 17
(26)
Here, D contains the strength of the interaction with the environment and describes the strength of localization. The term in (26) containing D can arise from a high-temperature environment, as it was discussed for example in [25], but it does not have to. It actually follows from a much wider class of situations [30]. Even if the environment is thermal, the temperature does not need to be high; in a well-known example calculated in [30] a small dust particle in intergalactic space is localized (decohered) by the ubiquitous microwave background radiation, a process described by an equation of the form (26). This master equation is also obeyed by the density matrix for the primordial fluctuations, which is discussed in [8]. We shall here consider the situation where a general initial state (not necessarily Gaussian) is present. In such a situation the Wigner function is usually not positive definite (cf. [48] for the realization of such states in the laboratory). The emergence of positivity is usually connected with decoherence, so it can be used alternatively to the density matrix as a measure for classicality. It was found in [47] that the Wigner function for the case (26) becomes positive after a certain time td , independent of the initial state. This, therefore, signals the emergence of classical behaviour. Moreover, it was shown there that the reduced density operator ρˆ can for t > td be decomposed in the form, Z ρˆ = dΓ P (Γ, t)|ΓihΓ| , P (Γ, t) ≥ 0 , (27) where |Γi denotes a set of Gaussian states. They play the role of the pointer states, so (27) is not the standard orthogonalization of the density matrix; note that the decomposition (27) is with respect to an overcomplete basis. Localization leads here to the emergence of narrow wave packets in position space (but not to delta functions). The width of these narrow Gaussians was determined by the ‘predictability sieve’ [24]. Apart from numerical factors of order one, the width is given by the expression δ ≡ (~/mD)1/4 . If the kinetic term in (26) is neglected compared to the localization term, we have m → ∞ and thus δ → 0 – the width of the Gaussian pointer states becomes very small, but they remain of course normalizable. For a small kinetic term the pointer states are thus approximate position eigenstates. This is the situation that has its analogue in the long-wavelength modes in cosmology. The limit m → ∞ is analogous to the limit r → ∞. This can be seen from the correlation p ≈ (tan ϕ)y ≈ e−r y, which holds for exponential inflation and expresses the smallness of the kinetic (p2 -containing) term. We shall address below the question of positivity for the Wigner function and the ensuing decoherence time in the cosmological case. While for the special initial state (3) the Wigner function is always positive, the question 18
of positivity is non-trivial for all other initial states, such as the ones in [21] (coherent states, states with an arbitrary number N of particles). The results of [47] were generalized in [46] to a system Hamiltonian of arbitrary quadratic form and to a general Lindblad equation. It was shown in particular that the Wigner function has always (apart from the case of Gaussian states) negative parts for t < td . The emergence of positivity at time td (and not earlier) is thus independent of the initial state. It was generalized to non-Markovian situations in [49]. It can thus be assumed that in a generic situation of a particle coupled to an environment, the state of the system can after a finite time not be distinguished from an exact mixture of particular Gaussian states which have a narrow width in case of strong interaction.
3.2
Lindblad equation for the primordial fluctuations
We shall restrict ourselves in the following to the case of one Lindblad operˆ The generalization to several Lindblad operators is straightforward ator L. [5]. Such a generalization is needed if one wants to accommodate non-linear effects by the primordial modes themselves, that is, to express the influence of ‘environmental’ k-modes on ‘system’ k-modes. Such a master equation was used in [18, 19], whereby the ‘system modes’ were chosen to be the longwavelength (observable) modes and the ‘environmental modes’ were chosen to be the short-wavelength (unobservable) modes with wavelengths smaller or of the order of the Hubble radius. The master equation then reads [5] dˆ ρ ˆ † Lˆ ˆ ρ − 1 ρˆL ˆ ρˆ] + Lˆ ˆ ρL ˆ† − 1 L ˆ†L ˆ. = −i[H, dt 2 2
(28)
We further assume as in [46] that the Lindblad operator is linear in our variables p and y. The general system Hamiltonian discussed in [46] is of the form ˆ = H11 p2 + 2H12 py + H22 y 2 . H (29) To conform with the conventions used there, we will from now on work with a slightly different set√of canonical variables, defined by the substitutions √ p → k p and y → y/ k in (1), so that the Hamiltonian reads a′ k 2 2 ˆ H= p + y + yp . 2 a
(30)
In our case we thus have H11 = H22 = k/2, and H12 = a′ /2a ≡ H/2, where H ≡ aH denotes the conformal Hubble parameter. In the analysis of [46], the 19
determinant of the quadratic form defined by (29) plays a crucial role. (We note that this analysis relies on a Hamiltonian that does not explicitly depend on time. Strictly speaking, therefore, our treatment is only valid for times that are reasonably short compared to the characteristic timescale given by H.) More precisely, the sign of this determinant decides about the qualitative features of the decoherence time. In our case, this determinant is positive for k > aH ≡ H (‘elliptic case’) and negative for k < aH (‘hyperbolic case’), that is, the change of sign just happens when the modes cross the Hubble radius. For modes outside the Hubble radius, the hyperbolic case is of relevance. This turns out to be of crucial importance. The ‘inverted oscillator’ is the paradigmatic example for the hyperbolic case [34]. Further following the notation used in [46], we write the Lindblad operator in the form p ′ ′′ , (31) L = (l + il ) · y where the real components of the ‘vectors’ ′′ ′ λ λ ′′ ′ and l = l = ′ µ′′ µ
(32)
contain all information on the interaction of the system with its environment. We also introduce coefficient’ α ≡ µ′ λ′′ − µ′′ λ′ as well as the √ √ the ‘dissipation quantity σ ≡ 2 − det H = H2 − k 2 . Note that σ is real for k ≤ H (outside the Hubble radius) and imaginary for k > H (inside the Hubble radius). The parameter α is non-vanishing if there is an energy exchange between system and environment. The real part of σ is called ‘Lyapunov exponent’ in [46], which is in accordance with our discussion of the entropy in [8]. It is nonvanishing only in the hyperbolic case. The quantum-mechanical example (26) is recovered from (28) in the special case of λ′ = λ′′ = 0, and one can then identify D = µ′2 + µ′′2 . This leads in particular to vanishing friction, α = 0. A quantitative result of [46], following [47], is that the positivity time, ηp , for the Wigner function is the solution of det M(−η) =
1 , 4
(33)
with a certain (time-dependent) matrix M that appears in the solution for the Wigner function and that depends on the Lindblad operators. For the
20
case of primordial fluctuations we obtain det M(−η) " # 2 1 (e2αη − 1) e4αη − 2e2αη cosh 2ση + 1 = A11 A22 − A12 A21 , (34) (4σk)2 α2 − σ 2 α2 where the Aij denote the components of a (complex, symmetric) matrix A that depends on the components of the Lindblad operator as well as on the system parameters H and k. For the reader’s convenience we give here the explicit expressions of the matrix elements, 2 2 kµ′ kµ′′ ′ 2 ′′ 2 A11 = k λ − +k λ − , H−σ H−σ 2 2 kµ′ kµ′′ ′ 2 ′′ 2 A22 = k λ − +k λ − , H+σ H+σ A12 = k 2 λ′2 + λ′′2 + µ′2 + µ′′2 − 2kH (λ′ µ′ + λ′′ µ′′ ) = A21 . While
A12 A21 =
2 ′2 k λ − 2kHλ′ µ′ + k 2 µ′2
+ k 2 λ′′2 − 2kHλ′′ µ′′ + k 2 µ′′2
2
(35)
is real and positive for a general non-vanishing Lindblad operator, A11 A22 = A12 A21 + (2αkσ)2 is real, but one cannot determine its sign easily for modes inside the horizon, where σ 2 < 0.
3.3
Modes outside the Hubble radius
Let us first treat a mode k of the cosmological perturbations outside the Hubble radius (σ 2 > 0), that is, before the second Hubble-radius crossing. Since the coupling to the environment can only lead to entanglement and not to disturbance in that situation, we take the dissipation coefficient α > 0 to be very small, that is, α ≪ |σ|. We note that for such modes, k2 k/H→0 −→ H . σ ≈H 1− 2 2H Since we have for inflation, a(η) = −
1 , (η − 2ηe )HI 21
we get ση ≈ 2Hηe − 1 → 1 at the end of inflation. Then, the condition (33) gives, using α ≪ σ, A212 cosh 2ση − 1 2 1≈ 2 2 , −η k σ 2σ 2 p yielding |ηp | ∼ k/|A12 |. Choosing λ′′ = 0 = µ′′ in order to implement α = 0, we have A12 ≈ k 2 (λ′ − µ′ )2 + 2λ′ µ′ k(k − H) ≈ −2λ′ µ′ kH ,
where in the last step we have assumed that the term with kH dominates. This then gives |ηp | ∼ |λ′ µ′ |−1 and, consequently, for the corresponding positivity time: H −1 |λ′µ′ | tp ∼ HI−1 ln I . (36) a0 At this moment, the entropy reaches the value HI−1 |λ′ µ′ | . a0 Note that the quantity tL ≡ a0 /|λ′ µ′ | is invariant under arbitrary rescaling of spatial coordinates and has the dimension of time. The most striking feature in this result is its approximate independence of the coupling parameters (they enter tp only logarithmically). In (13) the influence of the environment was modelled by the parameter ξ, effectively describing localization in the field-amplitude basis as one special case of interaction. So, the result just derived supports the earlier claim in [8] that the details of the coupling of primordial fluctuations to the environment are not important. The independence of the details on the coupling is a general feature of systems characterized by a ‘Lyapunov exponent’, independent of whether the system is chaotic or (as is the case here) just classically unstable [31]. The positivity time (36) is of the same order as the decoherence time discussed in [6, 7, 8]. Both times are generally related [47]. In this limit the density matrix can thus be decomposed into Gaussian pointer states according to (27). For strong coupling to the environment which leads to decoherence, these pointer states are narrow Gaussians with respect to the field amplitude. We note that concrete models in which a specific interaction in the action is chosen lead to results in accordance with our general expressions for decoherence time and entropy [15, 50]. We also want to comment on the situation far outside the Hubble radius, but in a radiation dominated universe. There the scale factor obeys 1/2 η t a(η) = = a , (37) e HI ηe2 te S ≈ HI tp ≈ ln
22
and thus one has again ση ≈ 1. Repeating the above calculations now yields for the decoherence time, tp ∼
HI a2e HI t2L = . 2(λ′ µ′ )2 2
(38)
For HI tL ≫ 1, where here tL = ae /|λ′µ′ |, this is a much longer time than the positivity time during inflation, which means that decoherence is here much less efficient than during inflation, in accordance with our earlier result [8]. The physical explanation of these results is the following. As was emphasized in [3], omission of the decaying mode is sufficient to get quantum decoherence. However, this mode should be really small to justify this procedure. In the super-Hubble limit, the only process making the decaying mode small is the expansion of the Universe; interaction with the environment destroys the initial correlation between decaying and growing modes but typically makes the amplitude of the decaying mode larger (that is reflected in the growth of the entropy S). During inflation, the expansion of the Universe is much faster (quasi-exponential) than during the radiation period (37). Thus, the decaying mode decreases much faster during inflation, which explains the difference between (36) and (38). Also, due to the exponential decrease of the decaying mode amplitude, coupling parameters describing real physical decoherence enter only logarithmically in tp . Note, however, that the fact that Htp in the second case is larger does not mean that the amount of entropy gained during decoherence is less in that case, too. This requires special investigation. It is also important that Htp & 1 in both cases. This means, in particular, that after the time period ∆t ∼ tp after the first Hubble radius crossing, the characteristic rms amplitude of the decaying mode of y is much less than the rms value of y in the initial vacuum state. This holds because the ratio of decaying to constant (‘growing’) mode is exp(−3H(t−tk )), and thus this ratio becomes small even for a only logarithmically large exponent. Therefore, the Wigner ellipse remains squeezed in one direction below its vacuum width even after the decoherence. This is another form of the result of Sec. 2 that S < Smax /2.
3.4
Modes inside the Hubble radius
Let us now examine the situation inside the Hubble radius. The great generality of (28) makes it hard to read off the qualitative features of the positivity time. (See, however, the argument in [46] that one only has to analyze one special example of the elliptic system in order to obtain its general qualitative 23
behaviour.) We want to consider a representative (and realistic) environment that is also used in [46]: a thermal bath of photons with average occupation number n. This calls for two Lindblad operators defined as p 0 α(n + 1) ′ ′′ p l1 = , l1 = , α(n + 1) 0 (39) √ − αn 0 ′′ ′ , l2 = l2 = √ . αn 0 The non-unitary part of (28), which is given by the dissipation coefficient α > 0, and the matrix A then become a sum over the different Lindblad operators. We then get 2 A11 A22 = [(2n + 1)2αkH]2 and A12 A21 = (2n + 1)2αk 2 . (40) The determinant (34) can now be written as det M(−η) [(2n + 1)k]2 = 4 |σ|2
2αη
e
−1
2
H2 4αη α2 2αη − e − 2e cos 2|σ|η + 1 α2 + |σ|2 k 2 (41)
with |σ|2 = k 2 − H2 . Again assuming the coupling to be small, α ≪ |σ|, the prefactor of the second term becomes much smaller than unity. If we exclude 2 the unrealistic case that e4αη − 2e2αη cos 2|σ|η + 1 ≫ (e2αη − 1) , the second term can safely be neglected to give 2 (2n + 1)k 2αη det M(−η) ≈ e −1 . (42) 2 |σ|
This yields a positivity time k≫H 1 |σ| 1 1 1 , ≈ ln 1 + ln 1 + ηp = 2α k 2n + 1 2α 2n + 1
(43)
where the last approximation is valid for modes long after the second Hubble radius crossing when |σ| ≈ k. (It can easily be checked that the above line of thought as well as the result follow from |σ| ≈ k alone.) Using Friedmann time, the positivity time (43) reads a2e 1 2 tp ≈ . (44) ln 1 + 16te α2 2n + 1 24
The positivity time is independent of the Hubble parameter H, but strongly depends on the (small) coupling α, in sharp contrast to the situation outside the Hubble radius. This is completely natural from the physical point of view since, inside the Hubble radius, there is no division of modes into growing and decaying ones: both linear independent solutions of the wave equation corresponding to the Hamiltonian (1) oscillate with amplitudes adiabatically decreasing with the expansion of the Universe; no more squeezing occurs. Thus, in this regime, the Universe expansion may not help us in getting decoherence, only real dissipative processes work in this direction. It is interesting to note that the limit n → 0 does not lead to tp /te → ∞, but rather to tp /te → (ln2 2)a2e /16te α2 . This does not seem to be an artifact of the approximations: it is due to the vacuum energy of the photon bath, and may vanish after proper renormalization. Other concrete models of interaction may use an interaction of the form V = hik φ,iφ,k where hik is a metric perturbation and plays the role of the system, while φ is an effective (‘phonon’) field describing small-scale perturbations of the thermal background and plays the role of the environment. Clearly, such an interaction results from the kinetic term of this field.
4
Conclusions and discussion
Using methods from the quantum theory of open systems, we have shown that primordial fluctuations decohere when their wavelength becomes much larger than the Hubble radius and that then the pointer basis is given by narrow Gaussians which approximate the field amplitude basis. Consequently, the entropy per mode is smaller than half the maximal entropy. Thus, the density matrix remains squeezed in one direction as compared to the pure vacuum state. However, this does not preclude the positivity of the Wigner function. If a particular interaction between the modes and other fields is given, one can calculate the Lindblad operators and all physical quantities in terms of this interaction and find the positivity time tp after which the Wigner function becomes positive everywhere and, therefore, may be approximated by a classical distribution in the phase space. In the super-Hubble regime, tp explicitly depends on the Hubble time H −1 and is during inflation only logarithmically larger than it. On the other hand, in the sub-Hubble regime after the second Hubble radius crossing, tp is independent of H and is totally determined by dissipation processes. As for the width, this leads to an expression analogous to (but more complicated than) the width (~/mD)1/4 for the localization of a particle. This crucial difference between decoherence mechanisms in the sub- and 25
super-Hubble regimes removes, we suppose, doubts regarding the possibility of (partial) decoherence during inflation expressed in the recent paper [19]. The answer to the question in this paper why we do not see similar decoherence in the Minkowski space-time is, first, because there is no super-Hubble regime in this case. Second, it was assumed in [19] that the ‘environmental’ (short-wavelength) modes are in their adiabatic ground state. Fields in their ground state are usually not able to exert a decohering influence [5]. So, the analysis presented here assumes that some fields are present which are not in their ground state. The concrete examples of such fields are cosmological perturbations themselves, scalar perturbations and gravitational waves, with scales of the order of the Hubble radius (the case not considered in [19]). But even if this was not true, the analysis in [3] has shown that it is impossible in practice to distinguish the squeezed state of the modes from a classical stochastic ensemble, cf. also the gedanken experiments discussed in [7, 26]. At last, it should be emphasized that we do not consider decoherence of generic long-wavelength modes but specifically only the decoherence of very strongly squeezed states. Then an exceedingly small interaction is already sufficient for a loss of quantum coherence, which has S & 1. This fact, namely that, even in the Minkowski space-time, strongly squeezed states are much more fragile than, for example, coherent states, is well known in standard quantum mechanics and represents the main obstacle to generate strongly squeezed states in the laboratory, see for example Sect. 3.3.3.1 in [5]. Still it should be noted that there is an agreement between our paper and [19] regarding the pointer basis in the super-Hubble regime. On the other hand, we would not fully support the statement of [18] that decoherence is “extremely effective” during inflation. What follows from our results is that decoherence, though quick and sufficient to reach the positivity of the Wigner function, is not sufficiently effective, for example, to make the Wigner ellipse exceeding its vacuum value in all directions (the latter would correspond to S > Smax /2).4 Thus, our results regarding the degree of decoherence reached during inflation are, in some sense, intermediate between those of [18] and [19]. This discussion shows how subtle is the problem of decoherence and quantum-to-classical transition for cosmological perturbations. 4
Note also that the ratio of inflaton gravitational to self-interaction is overestimated in [18]. One should take into account that, during inflation, the gravitational potential Φ describing scalar perturbations is not constant, but slowly growing from zero at the first Hubble radius crossing up to its final value after the end of quasi-exponential expansion of the Universe. Thus, its value during inflation is much less than that after inflation which can be observed now. Proper account of this fact makes the inflaton gravitational and self-interactions of the same order.
26
Acknowledgements C.K. is grateful to the Max Planck Institute for Gravitational Physics in Golm, Germany, for its kind hospitality while part of this work was done. He also thanks Robert Brandenberger, David Lyth, and Daniel Sudarsky for discussions. A.A.S. was partially supported by the Russian Foundation for Basic Research, grant 05-02-17450, and by the Research Program “Quantum Macrophysics” of the Russian Academy of Sciences, as well as by the German Science Foundation under grant 436 RUS 113/333/10-2. He also thanks the Centre Emile Borel, Institut Henri Poincar´e, Paris, for hospitality in the period when this paper was finished.
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